Overview of Monte Carlo Generators John Campbell, Fermilab Monte - - PowerPoint PPT Presentation

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Overview of Monte Carlo Generators John Campbell, Fermilab Monte - - PowerPoint PPT Presentation

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Overview of Monte Carlo Generators John Campbell, Fermilab Monte Carlo overview: history Theoretical description of a sample of events recorded by an experiment. Distinction between: flexible event generators that provide an

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John Campbell, Fermilab

Overview of Monte Carlo Generators

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Overview of MC Generators - John Campbell -

Monte Carlo overview: history

  • Theoretical description of a

sample of events recorded by an experiment.

  • Distinction between:
  • flexible event generators

that provide an exclusive description of the full final state

  • specialized parton level

predictions that can be systematically improved by including higher orders in perturbation theory

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e.g. Pythia, HERWIG e.g. MCFM, BlackHat

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Overview of MC Generators - John Campbell -

More recently ...

  • Difference significantly

blurred by an understanding

  • f how to combine exact

fixed order results with the event generator capability

  • f a parton shower
  • Better treatment of

hard radiation via merging of samples Systematic inclusion of next-to-leading order (NLO) effects

  • Will try to describe some of the

theoretical underpinning and general features of the above.

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e.g. ALPGEN, SHERPA e.g. POWHEG, (a)MC@NLO

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Overview of MC Generators - John Campbell -

Factorization

  • A theoretical description of the process relies on factorization of the problem

into long- and short-distance components:

  • Sensible description in theory only if scale Q2 is hard, i.e. production of a

massive object or a jet with large transverse momentum.

  • Asymptotic freedom then allows a perturbative expansion of all ingredients

that appear in the calculation, e.g.

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σAB =

  • dxadxb fa/A(xa, Q2)fb/B(xb, Q2) ˆ

σab→X

universal parton distribution functions hard scattering matrix element

ˆ σab→X = ˆ σ(0)

ab→X + αs(Q2)ˆ

σ(1)

ab→X + α2 s(Q2)ˆ

σ(2)

ab→X + . . .

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Overview of MC Generators - John Campbell -

Leading order

  • Simplest picture: leading order parton level prediction.
  • 1. Identify the leading-order partonic process that contributes to the hard

interaction producing X

  • each jet is replaced by a quark or gluon (“local parton-hadron duality”)
  • 2. Calculate the corresponding matrix elements.
  • 3. Combine with appropriate combinations of pdfs for initial-state partons a,b.
  • 4. Perform a numerical integration over the energy fractions xa, xb and the

phase-space for the final state X.

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usually a tree diagram, e.g. Drell-Yan ... but not always, e.g. Higgs from gluon fusion

σAB =

  • dxadxb fa/A(xa, Q2)fb/B(xb, Q2) ˆ

σ(0)

ab→X

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Overview of MC Generators - John Campbell -

Tools for LO calculations

  • Practically a solved problem for over a decade - many suitable tools available.
  • Computing power can still be an issue.
  • This is mostly because the number of Feynman diagrams entering the

amplitude calculation grows factorially with the number of external particles.

  • hence smart (recursive) methods to generate matrix elements.

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ALPGEN

  • M. L. Mangano et al.

http://alpgen.web.cern.ch/alpgen/

AMEGIC++

  • F. Krauss et al.

http://projects.hepforge.org/sherpa/dokuwiki/doku.php

CompHEP

  • E. Boos et al.

http://comphep.sinp.msu.ru/

HELAC

  • C. Papadopoulos, M. Worek

http://helac-phegas.web.cern.ch/helac-phegas/helac-phegas.html

MadGraph

  • F. Maltoni, T. Stelzer

http://madgraph.hep.uiuc.it/

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Overview of MC Generators - John Campbell -

Parton showers

  • A parton shower is a way of simulating the radiation of additional quarks and

gluons from the hard partons included in the matrix element.

  • This radiation is important for describing more detailed properties of events,

e.g. the structure of a jet.

  • Eventually, evolution produces partons that are are soft ( ~GeV) and that must

be arranged into hadrons (hadronization) → true event generator.

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evolution of parton shower hard parton from matrix element progressively softer partons

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Overview of MC Generators - John Campbell -

Underlying theory of parton showers

  • The construction of a parton shower is based on another type of factorization:
  • f cross sections in soft and collinear limits.
  • Easiest way to see the behavior is in the matrix elements.
  • In the limit that quark 2 and gluon 3 are collinear,

there is a remarkable factorization:

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p1 p1 p2 p2 p3 Q Q

virtual photon (Q2>0)

Pqq(z) = CF 1 + z2 1 − z

  • (2 diagrams)

p2 = zP , p3 = (1 − z)P |Mγ∗ ¯

qqg|2

coll.

− → 2g2

s

2p2.p3 |Mγ∗ ¯

qq|2Pqq(z)

|Mγ∗ ¯

qqg|2 = 8NcCF e2 qg2 s

(2p1.p3)2 + (2p2.p3)2 + 2Q2(2p1.p2) 4 p1.p3 p2.p3

  • |Mγ∗ ¯

qq|2 = 4Nce2 qQ2

splitting function

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Overview of MC Generators - John Campbell -

Universal soft and collinear factorization

  • The important feature is that this behaviour is universal, i.e. it applies to the

appropriate collinear limits in all processes involving QCD radiation.

  • They are a feature of the QCD interactions themselves.

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a c b z 1-z |Mac...|2

a, c coll.

− → 2g2

s

2pa.pc |Mb...|2Pab(z) Pqq(z) = CF 1 + z2 1 − z

  • Pqg(z) = TR
  • z2 + (1 − z)2

Pgg(z) = 2Nc z2 + (1 − z)2 + z2(1 − z)2 z(1 − z)

  • collinear singularity

additional soft singularity as z→1 soft for z→0, z→1

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  • Overview of MC Generators - John Campbell -

Parton showers

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dσn+1 = dσn αs 2π dt t Pab(z) dz

  • Factorization extends to the phase space too and hence to the level of cross

sections:

  • In this equation the virtuality, t is the evolution variable;
  • ther choices are angular ordered and pT ordered.
  • Here we have considered timelike branching (all particles are outgoing, t>0).
  • extension to the spacelike case (radiation on an incoming line) is similar.
  • This is the principle upon which all parton shower simulations are based.

t z 1-z dt t = dθ2 θ2 = dp2

T

p2

T

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Overview of MC Generators - John Campbell -

Sudakov logarithms

  • Solution of evolution equation in terms of Sudakov form factor,

corresponding to probability of no resolvable emission

  • Resolvable means not arbitrarily soft: remove singularities as z→0 and z→1:
  • Probability of no resolvable branchings from a quark:
  • Exponentiation sums all terms with greatest number of logs per power of αs

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t′ t0 < z < 1 − t′ t0 ∼ exp

  • −CF

αs 2π t

t0

dt′ t′ 1−t0/t′

t0/t′

dz 1 − z

  • ∆q(t) = exp
  • −

t

t0

dt′ t′ 1−t0/t′

t0/t′

dz αs 2π

  • Pqq(z)
  • ∼ exp
  • −CF

αs 2π

  • log2 t

t0

  • leading log

parton shower

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Overview of MC Generators - John Campbell -

Hadronization

  • At very small scales of t perturbation theory is no longer valid
  • no further branching beyond ~ GeV.
  • All partons produced in the shower are showered further, until same condition.

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  • Once this point is reached, no more

perturbative evolution possible.

  • Partons should be interpreted as

hadrons according to a hadronization model.

  • examples: string model (Pythia),

cluster model (Herwig, Sherpa).

HADRONIZATION partonic matrix element parton shower

  • Most importantly: these are phenomenological models.
  • They require inputs that cannot be predicted from the QCD Lagrangian ab

initio and must therefore be tuned by comparison with data (mostly LEP).

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Overview of MC Generators - John Campbell -

Popular parton shower programs

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HERWIG

  • G. Corcella et al.

http://hepwww.rl.ac.uk/theory/seymour/herwig/

HERWIG++

  • S. Gieseke et al.

http://projects.hepforge.org/herwig/

SHERPA

  • F. Krauss et al.

http://projects.hepforge.org/sherpa/dokuwiki/doku.php

ISAJET

  • H. Baer et al.

http://www.nhn.ou.edu/~isajet/

PYTHIA

  • T. Sjöstrand et al.

http://home.thep.lu.se/~torbjorn/Pythia.html

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Warnings

  • By construction, a parton shower is correct only for successive branchings that

are collinear or soft (i.e. only leading logs, or next-to-leading logs with care).

  • Must take care when

describing final states in which there is either manifestly multiple hard radiation, or its effects might be important.

  • example: simulation of

background to a SUSY search in the ATLAS TDR.

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PS (bkg) SUSY signal improved background calculation

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Parton shower extensions

  • As simplest example, consider Drell-Yan process:

(one power of αs per jet)

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accuracy of NLL parton shower accuracy of tree-level Z+2 jet calculation

How can parton shower recover more of fixed-order accuracy?

pp → Z (+n jets)

c.f. earlier, leading log: αn

s L2n 2 4 6 8

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Tree-level matching

  • Use exact matrix elements for the hardest emission from the parton shower

instead of approximate form.

  • Captures one extra term in the

expansion

  • does not account for all corrections
  • real radiation is taken into account

but not virtual (loop diagram) contributions

  • Hence shape improved for observables

dominated by low-multiplicity emission, but overall normalization same as before.

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PS + one matched emission

Mostly historical, not a feature of modern generators

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Multi-jet merging in pictures

  • Merging: include more exact matrix elements as initial hard scatters,

with merging scale determining transition from approximate to exact MEs.

17 hard final state

+ + + ... + ... + + ...

usual shower + 1-jet merging + 2-jet merging

shower (approx) matrix element (exact)

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Multi-jet merging

  • Perform matrix element corrections to multiple emissions
  • Introduces an unphysical merging scale in order

to perform corrections for each jet multiplicity

  • again, impact only on shapes of

relevant distributions - for observables up to the number of jet samples merged

  • again, no possible improvement in rate

cross section remains a leading order estimate

  • Various techniques for combining samples

without overcounting in shower:

  • CKKW (Catani, Krauss, Kuhn, Webber)

CKKW-L (Lönnblad)

  • MLM (Mangano)

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PS + multi-jet merging for up to 3 jets

SHERPA ALPGEN

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Overview of MC Generators - John Campbell -

Higher orders

  • Systematic method of improving the perturbative prediction
  • at each order of calculation, result is formally independent of the choice

made for the renormalization scale (used in strong coupling) and factorization scale (used to evaluate pdfs)

  • uncertainty is usually estimated by examining residual sensitivity to these

scales; typically reduced as more orders are considered

  • scales should be motivated by physics; range of variation is a subject for

debate and resulting uncertainties certainly not Gaussian!

  • Approximate hierarchy:
  • LO: ballpark estimate (~ within a factor of two)
  • NLO: first serious estimate (~ 10% accuracy)
  • NNLO: precision (~ few percent), serious estimate of uncertainty

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General structure of NLO calculation

  • Two divergent contributions, only finite together (Kinoshita-Lee-Nauenberg)

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+

real radiation virtual radiation (loop)

  • counter-

terms

+

counter- terms

soft/collinear singularities cancelled numerically singularities cancelled analytically

  • In general: many “counter-events” for each real radiation event.
  • Naive “event generator” produces mixed sample of parton configurations with

large positive and negative weights; not suitable for usual analysis.

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Parton-level NLO

  • Automated 1-loop approaches
  • great flexibility and range of applications; heavy use of

numerical methods; may be slow

  • MadLoop: mostly limited by CPU time
  • HELAC-NLO: e.g. tt+2 jets, tttt
  • GoSam: e.g. WW+2 jets, H+3 jets
  • Specialized codes
  • less flexibility; may contain more sophisticated treatments

and much faster

  • MCFM: W/Z/H+2 jets, Wbb, Zbb, diboson, top processes
  • VBFNLO: double and triple boson production, VBF
  • Rocket/MCFM+: W+3 jets, WW+up to 2 jets
  • BlackHat: 4 jets, W/Z + up to 5 jets, photon+jets

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Cullen et al Bevilacqua et al JC, Ellis, et al Arnold et al Bern et al Melia et al Alwall et al

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NNLO progress

  • Even more diagrams and singularities to consider at NNLO
  • beginning of era of widespread availability of NNLO: W, Z, H (for a while),

diphoton, top pairs, Higgs+jet, jet production (in the last couple of years)

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Barnreuther, Czakon, Mitov

decreasing error bands: LO NLO NNLO NNLO + NNLL (resummation)

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NLO + parton shower

  • Many NLO parton-level predictions available -

but most come without parton shower benefits.

  • do not produce unweighted events
  • hard to correct for detector effects
  • Obvious problem:
  • NLO already includes one extra parton emission.
  • the hard part of this can be matched as before.
  • the soft/collinear part contains singularities that

must be accounted for in the usual way

  • Technical challenge now overcome.

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PS + NLO inclusive

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NLO + parton shower overlap

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+

singular radiation virtual radiation (loop)

  • counter-

terms

+

counter- terms

NLO

hard final state

+ + + ...

parton shower

problematic

  • verlap
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Solution

  • Two main methods for dealing with the overlap: POWHEG and MC@NLO.
  • although formally of the same accuracy, differences do occur

(see Nason and Webber, arXiv:1202.1251)

  • POWHEG (Frixione, Nason, Oleari)
  • “local K-factor”, real exponentiated in Sudakov, positive-weight generator
  • many processes implemented in POWHEG-BOX; same overall framework

for each process, but many contributions from different theorists

  • MC@NLO (Frixione, Webber)
  • usual Sudakov factor but possibility of negative weights
  • procedure has been automated in aMC@NLO (Alwall et al)
  • access to NLO+parton shower predictions at unprecedented level

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NLO+PS in action

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Frixione and Webber (2003) transverse momentum of top pairs, from MC@NLO differences in regions where NNLO important

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SHERPA

  • POWHEG and MC@NLO methods also used in SHERPA.
  • Recent application to W+1,2,3 jets.
  • Smooth interpolation between POWHEG, MC@NLO procedures.

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Höche, Krauss, Schönherr, Siegert

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Beyond MC@NLO and POWHEG

  • NLO inclusive cross section, exact matrix elements for further jets.
  • “MENLOPS” and “ME&TS”.
  • NLO precision for each jet emission:

merging samples that are each matched to correct NLO

  • “MEPS@NLO”
  • Many recent developments

and ongoing intense activity.

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Hamilton, Nason; Höche, Krauss, Schönherr, Siegert Höche, Krauss, Schönherr, Siegert

MENLOPS: W+0 jet NLO, W+1,2,3,4 LO MEPS@NLO: W+0,1,2 jets NLO, W+3,4 LO

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Crystal ball

  • Next frontier: parton shower + NNLO?
  • problem: NNLO calculations very slow and numerically delicate
  • will get there eventually, but cannot reasonably expect much progress in the

next few years

  • Orthogonal direction: improvements in the parton shower evolution
  • example: treatment of color,

where parton showers usually work in the “leading color approximation”

  • also called “planar”;

gluon = (quark, antiquark) in terms of color.

  • will eventually need this level
  • f precision too

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i j k l

planar QED-like

tA

ijtA kl = 1

2

  • δilδjk − 1

N δijδkl

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Summary

  • At the parton level:
  • NLO calculations standard, mostly available in an automated form
  • more complicated NLO calculations available in standalone code
  • NNLO calculations becoming available for an array of very important cases
  • Modern parton showers come in many flavors:
  • capability for multi-jet merged samples (MLM, CKKW)
  • NLO matched to first emission (MC@NLO, POWHEG)
  • NLO matched to first emission, multi-jet merged (ME&TS, MENLOPS)
  • NLO matched for many jets (MEPS@NLO)
  • Availability and maturity of predictions in that order

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